Delightful problem indeed! (And I am adding it to my list of useful ideas/concepts/problems for students learning math - and indeed for all those interested in 'learning to think').

I had vaguely guessed that the answer might be something like using the line connecting the centres of the two rectangles - but I must confess I looked at the answer and did not work it out or sketch it.

GSC Richard Strausz posted Dec 18, 2012 3:20 PM: > This is one of my favorites. I didn't figure it out> before seeing the solution...> > Richard> ============> > RAY: My neighbor Joan decides to bake some brownies> for her two little grandsons. She got one of those> rectangular pans. Because these kids are really> competitive, she knows she has to divide what she> bakes right in half -- which is pretty easy to do if> you've got a rectangular pan.> > So, she bakes the brownies, takes them out and puts> them on the cooling rack. Before she cuts it in half,> however, her husband comes along and cuts a rectangle> out of the middle, at random.> > Imagine, now, you've got a rectangular cake, and he> cuts a rectangle out of the middle. He wasn't even> nice enough to cut it out of the corner! The cuts> aren't even parallel to the sides of the original> cake.> > It's not touching an edge, but it could be, and it's> not necessarily parallel to any of the sides.> > She says, "What's this all about? How am I going to> cut it in half now?"> > He says, "Well, I guess you'll just have to bake more> brownies, and I'll eat this."> > She was bemoaning this to one of her girlfriends on> the phone, and the girlfriend says, "I have a remedy> for your dilemma." Joan asks, "Does it involve rat> poison?"> > The girlfriend says, "No, I have a way that you can> cut this cake in half and make sure that each of your> grandchildren gets the same amount of brownie. In> fact, I have two ways to do it. There's a hard way> and an easy way."> > Joan says, "Give me both ways."> > So, here's the question. How can you, with one cut of> the knife, cut the brownies in half?> ============> RAY: Here's the answer. The hard way is to take your> knife, and holding it parallel to the cooling rack> that the brownie is sitting on, slice the brownies> that way. One kid's going to get the top half of the> brownie and the other kid is going to get the bottom> half. I don't like that solution because the top half> and the bottom half are not truly equal.> > And here's the other solution. If you take a> rectangle, how would you find the center of a> rectangle? You would draw diagonals and you would> find the center.> > Any line drawn through the center of a rectangle,> that's not a diagonal, also cuts the rectangle in> half. So if you were to draw a line through the> center of the big rectangle and it went through the> center of the hole, then you would cut the brownie> into exactly two, equal pieces.> > So what you do is find the center of the little> rectangle that her husband cut out, by making two> diagonals, then you draw two diagonals for the big> piece of the brownie.> > It makes no difference where the other rectangle is,> if you connect the two centers with a straight line,> and continue right through to the edges. You will> have cut the brownie in half.